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M.-How, then, will you call this solid?
P.-Dodecahedron.

M.-How will you distinguish this dodecahedron from the former ?

P.--This may be called rhombic or rhomboidal dodecahedron; the former, pentagonal dodecahedron. M.-How many plane angles are there about each of its solid angles?

P.-About some of its solid angles there are three plane angles; about others, four.

M.-Compare these angles. Do you observe some difference respecting them?

P.-Those of which there are three about a solid angle are greater than those of which there are four. M.-Define these angles more strictly.

P.-Some of the solid angles are formed by three obtuse angles; others, by four acute angles.

M.-How many of the angles of a rhomb are obtuse? how many are acute?

P.-In a rhomb there are two obtuse and two acute

angles.

M.-How are they situated?

P.-They are opposite each other.

M.-How many obtuse angles are there in all the faces of the rhomboidal dodecahedron, and how many of them are acute angles?

P.-There are twenty-four obtuse and twenty-four acute angles in all the faces.

M.-How many solid angles, then, has this solid? and how many of each sort?

P.-There must be eight solid angles formed by the obtuse angles, and six solid angles formed by the acute angles; in all, fourteen solid angles.

M.-See whether it is so.

By what name will you

distinguish each sort of solid angles?

P.-Those formed by the obtuse angles may be called obtuse solid angles; the others, acute.

M.-How many edges has this solid?

P.-Twelve edges.

This and some of the following lessons being dependant on the results of the preceding, they may be considered as a repetition and application of the former. It is therefore not necessary to cause the substance of these lessons to be committed to memory.

LESSON VII.

THE BIPYRAMIDAL DODECAHEDRON.

M.-Examine this solid. What are the faces, and what is their number?

P.-The faces are isosceles triangles; their number is twelve.

M.-How will you call this solid?

P.-Dodecahedron.

M.-By what name will you distinguish this dodecahedron from the other two? It is usually called bipyramidal dodecahedron, from another kind of solids with which you will hereafter become acquainted.Examine its solid angles.

P.- Some are formed by six plane angles; others, by four.

M.-Endeavour to distinguish these angles the one from the other.

P.-Some of the solid angles are formed by four angles which are at the bases of the isosceles triangles: the other solid angles are formed by the angles which are opposite to the bases of the isosceles triangles.

M.-How many angles are there at each of these bases of the triangles ?

P.-Two angles at each base.

M.-How may are there of this sort in all the faces? P.-Twenty-four.

M.-How many solid angles formed by the angles at the base, then, has this solid?

P.-Six solid angles.

M.-And how many solid angles formed by the angles which are opposite the base ?

P.-Two solid angles.

M.—What, then, is the total number of solid angles of the bipyramidal dodecahedron, and what must be the number of its edges?

P.-It has eight solid angles, and eighteen edges. M.-See whether it is so.

THE TRAPEZOHEDRON.

M.-Examine this solid. What are its faces?

P-Its faces are quadrilateral figures-they are trapeziums,

M.-This solid is therefore called trapezohedron. What is the number of its faces?

P.-It is bounded by twenty-four trapeziums. M.-Examine the angles of each trapezium. P.-Three of the angles are acute angles, the fourth is obtuse.

M.-Examine the solid angles.

P.-Some of them are formed by four acute, the others by three obtuse angles.

M.-Calculate the number of solid angles of each sort, the total number of solid angles, and the number of edges.

P.-It has eighteen acute solid angles, eight obtuse solid angles; the total number is twenty-six solid angles, and the number of edges is forty-eight. M.-See whether it is so.

LESSON VIII.

THE PYRAMID.

M. (putting several pyramids before his pupils.)— Examine these solids. What do you observe in

them?

P.-The faces of each are triangles, except one, which is either a quadrilateral figure, or a pentagon, hexagon, &c.

M.-These kind of solids are called pyramids. How would you call the unequal face?

P.-The base of the pyramid.

M.—What kind of face may the base of a pyramid

be?

P-Any polygon whatever.

M.-How are the triangles situated?

P. The triangles all meet in one point, which is opposite the base of the pyramid.

M.-That point is called the summit of the pyramid. What is the number of triangular faces?

P.-There are as many triangular faces as the base has sides.

M.—What, then, is the number of faces of a pyramid ?

P. As many faces as the base has sides, more one. M.-Examine the solid angles.

P.-The angles at the base of the pyramid are formed each by three plane angles; and the angle at the summit, by as many plane angles as there'are triangles.

M.-What kind of triangles are they?

P.-Isosceles triangles.

M.-What is the number of solid angles in each pyramid?

P.-There are as many solid angles as the base has sides, more one.

M.-If the base of a pyramid is a triangle, by what name will you distinguish it from another whose base is a quadrilateral figure?

P.-The former may be called a triangular pyramid; the latter, quadrangular.

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